Degrees

9 vertices of degree 3, 11 vertices of degree 4, 10 vertices of degree 5.

Correct

Such a graph would violate Handshaking Lemma: the sum of degrees here is 121, which is odd.

2 vertices of degree 3 and 3 vertices of degree 2.

1 vertex of degree 1, 2 vertices of degree 2, 1 vertex of degree 3, 2 vertices of degree 5.

Correct

Though Handshaking Lemma is satisfied (1+2+2+3+5+5=18 is even), there is no graph with such vertex degrees. (We do not consider multigraphs or pseudographs here.) The reason is as follows: the total number of vertices is 6, thus the two vertices of degree 5 should be connected to all other vertices. Contradiction: the vertex of degree 1 could not be connected to two vertices.

11

Correct

The number of edges in a graph (games played) is counted by summing degrees of vertices: 6+5+3+3+2+2+1=22 and dividing by 2.

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